Retracts of Products of Chordal Graphs

Abstract : In this article, we characterize the graphs $G$ that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain $K_{2, 3}$, the $4$-wheel minus one spoke $W_4^{-}$, and the $k$-wheels $W_k$ (for $k\geq 4$) as induced subgraphs. We also show that these graphs $G$ are exactly the cage-amalgamation graphs as introduced by Brešar and Tepeh Horvat (Cage-amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071–1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of $G$ by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed with an intrinsic Euclidean metric, is a CAT(0) space. This generalizes similar results about median graphs as retracts of hypercubes (products of edges) and median graphs as 1-skeletons of CAT(0) cubical complexes.
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Submitted on : Monday, September 7, 2015 - 4:30:34 PM
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Bostjan Bresar, Jérémie Chalopin, Victor Chepoi, Matjaz Kovse, Arnaud Labourel, et al.. Retracts of Products of Chordal Graphs. Journal of Graph Theory, Wiley, 2013, 73, pp.1616180. ⟨10.1002/jgt.21665⟩. ⟨hal-01194854⟩



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